1. 2.1 Inductive Reasoning and Conjecture 2.2 Logic 2.3 Conditional Statements 2.8 Angle Relationships Chapter 2: Reasoning & Proof

2. 2.1 Inductive Reasoning & Conjecture Objective: We will be able to make conjectures based on inductive reasoning and find counter-examples. Logic and reasoning are used throughout geometry ______________________________________.to solve problems and reach conclusions Conjecture: ____________________________________________________________ Inductive Reasoning: ___________________________________________________ _______________________________________________. An educated guess based on analyzing info. or observing a pattern. Using a number of specific examples to arrive at a generalization or prediction 136 1015 21 I make a ______________ that the next number will be _______, My _______________________ is that each time the number increases by _____ more. Example: inductive reasoning L 1 For points L, M, and N: LM = 20, MN = 6, and LN = 14 Make a conjecture and draw a figure to illustrate Conjecture: L, M, and N are collinear Counterexample: ________________A false example It takes only one false to show a conjecture is not true. +2+3 +4 +5 +6 M N 14 6 20 Examples: Try Check for Understanding p. #4 - 9 4.4. 5. -8, -5, -2, 1, 4, ___ conjecture 21

3. 2.2 Logic Objective: We will be able to determine truth values of conjunctions & disjunctions and construct truth tables. How does logic apply on tests at school ? __________________________Answering T/F questions! Statement: ________________________________________________. Truth value: ______________________________________. A sentence that is either true or false, but not both The truth or falsity of a statement (T/F) I was born in St. Louis. statement Negation of a statement: _________________________________________________. Has the opposite meaning as well as opposite truth value. Summary representation example truth value negation use a lower case p or q or rp : I own three cats. T F I do not have five brothers. Boston is a city in MA. Conjunction Disjunction Compound statement joining 2 or more statements with the word: ___________ Remember: a conjecture is an educated guess. “p and q” “p or q” p is true, AND q is true (true, ONLY when both statements are true) AND OR p is true, OR q is true (true, IF AT LEAST one of the statements is true) I have five brothers. I was born in St. LouisandFALSE

4. Example 1: Determining Truth Values and Basic: truth value a) p and q: __________________________________________ p F Write the conjunction or disjunction using the statements, and then determine its truth value. Conjunctions Disjunctions January 1 st is the first day of the year. : ____ A convenient method for organizing the truth values of statements. p : T and q : F so p p Truth Tables - 5 + 11 = - 6 : ____ A triangle has three sides. : ____ q r January 1 st is the first day of the year - 5 + 11 = - 6. A triangle has three sides and January 1 st is the first day of the year. c) p and not r: _____________________________________________ e) p or r: ______________________________________________ January 1 st is the first day of the year and or a triangle does not have three sides. a triangle has three sides. and or a triangle has three sides. a triangle does not have three sides. T r :T and p :T so F T T T T F F T Conjunction q F F T T F T T F F F T F p Disjunction q F F T T F T T F F T T T

5. Example 2: Constructing truth tables 1. Make columns with headings (start with p, q, and r.) p q F F T T F T T F T F F T 2. List the possible combinations for the statements (use patterns!). 3. Use the truth values of the statements to determine any of the values of the “not” (opposite) statements. 4. Consider any conjunctions/disjunctions (start in parentheses first) F F F T p q F F T T F T T F T F F T F F F r F F T T T T F F T T T T F F F F F F T T T T T T F T T T T T F F F First pattern: first half T, second half F Second pattern: alternate T and F Third pattern: two T, two F, repeat… Suggestion: In order to create all possible combinations use the following pattern:

6. 2.3 Conditional Statements Objective: We will be able to analyze statements in if-then form and write the converse, inverse, and contra-positive. Conditional statement – a statement that can be written in if-then form Example 1: (underline hypothesis. ; circle the conclusion) If points A, B, and C lie on line l, then they are collinear. (Do NOT include the words if and then when writing the hypothesis and conclusion.) hypothesis The Bears will play in the tournament if they win their next game. “ if p, then q” conclusion If __________________, then ____________________. If ______________________, then ______________________________. they win their next game the Bears will play in the tournament Example 2: Example 3: If you are an NBA basketball player, then you are at least 5’6” tall. Hypothesis: _______________________________________ Conclusion: _______________________________________ Example 4: (Identify the hypothesis and conclusion of each statement. Then write each statement in if-then form.) (or “p implies q”) Hypothesis: ______________________________ Conclusion: ______________________________ If-then form: ______________________________ Adjacent angles have a common vertex. Hypothesis: ______________________________ Conclusion: ______________________________ A five-sided polygon is a pentagon. If-then form: ______________________________ ______________________________ you are an NBA basketball player you are at least 5’6” tall If two angles are adjacent, then they have a common vertex. two angles are adjacent they have a common vertex If a polygon is five-sided, then it is a pentagon. a polygon is five sided it is a pentagon Ex. “If you buy a new car, then you will get a free GPS system.”

7. Truth Values of Conditionals Hypotheses, conclusions, as well as conditional statements can have a truth value. Example: Determine the truth value of the following statement for each set of conditions. If it does not rain this Saturday, we will play football. given hypothesis and conclusion p q F F T T F T T F T T T F statement formed by conditional inverse contrapositive symbols example converse exchange the hypo. & concl. of the conditional negating both the hypo. & concl. of the conditional negating both the hypo. & concl. of the converse If two angles have the same measure, then they are congruent. If two angles are congruent, then they have the same measure. If two angles are not congruent, then they do not have the same measure. If two angles do not have the same measure, then they are not congruent. a) It does not rain this Saturday, and we will play football. b) It does not rain this Saturday, and we will not play football. c) It does rain this Saturday, and we will play football. d) It does rain this Saturday, and we will not play football. The hypothesis is true and the conclusion is true which makes the conditional statement true. Related Conditionals: The hypothesis is true, but the conclusion is false. Because the result is not what was promised, the conditional statement is false. The hypothesis is false, but the conclusion is true, but the statement does not say what you will do if it does rain. You could still play, so in this case, we cannot say the conditional is false. The hypothesis is false, but again, the statement does not say what you will do if it does rain so we cannot say the conditional is false.

8. Example: Related Conditionals Write the converse, inverse, and contrapositive of the statement, “All squares are quadrilaterals.” Determine whether each statement is true or false. If a statement is false, give a counterexample. First, write the conditional in if-then form. Conditional: ____________________________________ The conditional is TRUE If a figure is a square, then it is a quadrilateral. Write the converse by switching the hypothesis and the conclusion of the conditional Converse: _____________________________________ If a figure is a quadrilateral, then it is a square Keep the original statement and negate both parts. The contrapositive is the negation of the hypothesis and conclusion of the converse. Inverse: _________________________________________ Contrapositive: _________________________________________ If a figure is not a square, then it is not a quadrilateral. If a figure is not a quadrilateral, then it is not a square. The contrapositive is TRUE The converse is FALSE. The inverse is FALSE. Example: Linear pairs of angles are supplementary. Write the converse, inverse, and contrapositive of the statement. Determine whether each statement is true or false. If a statement is false, give a counterexample. Inverse: ______________________________________________________________ Contrapositive: ____________________________________________________________ Converse: _________________________________________________________ Conditional: ______________________________________________________ Truth Value If two angles form a linear pair, then they are supplementary. If two angles do not form a linear pair, then they are not supplementary. If two angles are supplementary, then they form a linear pair. If two angles are not supplementary, then they do not form a linear pair. True * Watch order below! False

9. 2.8 Angle Relationships (Review) Complementary Angles:Supplementary Angles: Vertical Angles:Adjacent Angles:Linear Pair: Perpendicular Lines: 1 2 1 2 1 2 1 2 1 2 1 2 3 4 Share a common side Can be comp. or suppl. Can be a linear pair Their non-common sides are opposite rays Angle Bisector: 1 2 Equilateral Triangle: Isosceles Triangle: 3 equal side lengths & 3 equal angles 2 equal side lengths & 2 equal angles

10. Share a common side Can be comp. or suppl. Their non-common sides are opposite rays 3 equal side lengths & 3 equal angles 2 equal side lengths & 2 equal angles Can be a linear pair 1 2 1 2 1 2 1 2 1 2 1 2 3 4 1 2

11. Examples! 1 2 1 2 1 2 1 2 3 6 7 8 1 2